Gromov–Witten theory of [ℂ2∕ℤn+1] × ℙ1

Zhou, Zijun; Zong, Zhengyu

doi:10.2140/ant.2022.16.1

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Gromov–Witten theory of $[\mathbb{C}^2 / \mathbb{Z}_{n+1}] \times \mathbb{P}^1$

Zijun Zhou and Zhengyu Zong

Vol. 16 (2022), No. 1, 1–58

DOI: 10.2140/ant.2022.16.1

Abstract

We compute the relative orbifold Gromov–Witten invariants of $[ℂ^{2} ∕ ℤ_{n + 1}] \times ℙ^{1}$ with respect to vertical fibers. Via a vanishing property of the Hurwitz–Hodge bundle, 2-point rubber invariants are calculated explicitly using Pixton’s formula for the double ramification cycle, and the orbifold quantum Riemann–Roch. As a result parallel to its crepant resolution counterpart for $𝒜_{n}$ , the GW/DT/Hilb/Sym correspondence is established for $[ℂ^{2} ∕ ℤ_{n + 1}]$ . The computation also implies the crepant resolution conjecture for the relative orbifold Gromov–Witten theory of $[ℂ^{2} ∕ ℤ_{n + 1}] \times ℙ^{1}$ .

Keywords

relative orbifold GW theory, GW/DT correspondence, crepant resolution conjecture

Mathematical Subject Classification 2010

Primary: 14N35

Milestones

Received: 1 February 2019

Revised: 1 March 2021

Accepted: 15 April 2021

Published: 22 February 2022

Authors

Zijun Zhou
Kavli Institute for the Physics and Mathematics of the Universe The University of Tokyo Kashiwa Japan

Zhengyu Zong
Department of Mathematical Sciences Tsinghua University Beijing China

Vol. 16, No. 1, 2022